4. Show that an arbitrary square matrix A can be written as where Ai is a symmetric matrix and A2 is a skew-symmetric matrix 4. Show that an arbitrary square matrix A can be written as where Ai...
A square matrix is called skew-symmetric if AT = -A. (a) (4 points) Explain why the main diagonal of a skew-symmetric matrix consists entirely of zeros. (b) (2 points) Provide examples of a 2 x 2 skew-symmetric matrix and a 3 x 3 skew-symmetric matrix. (6 points) Prove that if A and B are both n x n skew-symmetric matrices and c is a nonzero scalar, then A + B and cA are both skew-symmetric as well. (4 points) Find...
please explain in full details. A square matrix A is skew-symmetric if A = -A (a) If A is an n xn skew-symmetric matrix, with n odd, prove that A is singular, i.e. non-invertible (b) Find a skew-symmetric matrix that is invertible.
Let C be square matrix. i) Check if S = C + CT is symmetric ii) Check if N + C - CTis skew symmetric iii) Prove that every square matrix can be written as a sum of skew symmetric matrix and symmetric matrix
4. Let A be a square matrix such that AAT-1. Show that AI = ±1. 4. Let A be a square matrix such that AAT-1. Show that AI = ±1.
1. Writ A as the sum of Ma symmetric matrix and Na skew-symmetric matrix. 1 24 A=430= M+N 865 where, MT M and N-N. A+A Conclusion: M = and N= 2
R3. Problem 4. (3+2+3 pts) Consider an arbitrary skew-symmetric tensor 22: R3 (i) Show that I be represented by a vector w as follows: VE R3 Ωυ =ωXυ. (ii) Next, show that cofactor of 12 equals wow. (iii) Prove that 1+1 is always invertible.
We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be the set of all 2x2 skew-symmetric matrices: W = {A in m2x2(R) l A^T=-A}. (a) Show that W is a subspace of M2x2(R) (b) Find a basis for W and determine dim(W). (c) Suppose T: M2x2(R) is a linear transformation given by T(A)=A^T +A. Is T injective? Is T surjective? Why or why not? You do not need to verify that T is linear. 3. (17 points)...
2) If A is a symmetric matrix, show that the norm of A can be the maximum eigenvalue of A or the square root of the maximum eigenvalue of ATA. Explain this by proving that the maxim um eigenvalue of A the same as the square root of the maximum eigenvalue of A A when A is symmetric.
16. Let x and y be vectors in R3 and define the skew- symmetric matrix A, by 10-X3 X2 A = X3 0 -X1 I-X2 x 0 (a) Show that x x y = Axy. (b) Show that y x x = Amy.
7. Matrix A is said to be involutory if A2 = 1. Prove that a square matrix A is both orthogonal and involutory if and only if A is symmetric.